In　this　study,　we　address　a　new　consistent　vehicle　routing　problem　(ConVRP)　by　considering　driver　equity　and　flexible　route　consistency.　It　aims　to　optimize　the　departure　time　and　routes　for　a　set　of　vehicles　over　a　defined　planning　period,　aiming　to　minimize　the　total　travel　time　while　considering　the　discounts　for　the　route　consistent　segments.　Previous　studies　on　ConVRP　have　primarily　focused　on　ensuring　customer　satisfaction　through　time　consistency　and　driver　consistency,　while　overlooking　driver　equity.　In　our　ConVRP,　we　introduce　a　flexible　strategy　to　attain　route　consistency,　i.e.,　drivers　are　encouraged　to　traverse　familiar　routes　as　many　as　possible　during　the　planning　period　by　offering　time　discounts　for　routes　crossed　more　than　twice.　Driver　equity　is　addressed　through　the　optimal　allocation　of　delivery　capacity.　For　this　problem,　we　formulate　it　into　a　mixed-integer　linear　programming　model.　Given　its　strong　NP-hardness,　a　tailored　adaptive　large　neighborhood　search　algorithm　(ALNS)　is　developed　to　solve　practical-sized　problems.　Destroy　and　repair　operators　are　adaptively　applied　to　remove　customers　from　the　routes　and　reinsert　them　in　better　positions.　A　new　repair　operator　that　identifies　suitable　customer　locations　by　considering　vehicle　load　and　an　exchange　operator　that　reverses　part　of　the　specified　routes　are　proposed　to　obtain　better　solutions　and　increase　search　efficiency,　respectively.　Moreover,　we　enhance　arrival　time　consistency　by　adjusting　departure　time.　Extensive　numerical　experiments　for　instances　of　varying　scales　are　conducted　to　evaluate　the　proposed　algorithm.　Results　demonstrate　that　i)　the　proposed　algorithm　can　obtain　high-quality　solutions　(the　average　relative　difference　compared　to　the　optimal　solution　is　only　1.68%);　ii)　Our　approach　can　also　find　better　solutions　(with　an　average　improvement　rate　of　3.53%　and　33.70%,　respectively)　in　a　short　computation　time,　when　compared　to　the　basic　ALNS　without　a　new　operator　and　the　widely　used　variable　neighborhood　descent　algorithm;　and　iii)　setting　a　small　discount　allows　decision-makers　to　significantly　enhance　consistent　distance　for　drivers.　©　2023　Elsevier　Ltd